I am simulating bubble collapse near wall. when I assume compressible air inside the bubble and incompressible water, my simulation goes well and I get good results. But when I INTERPRET a udf for compressibility of water, I get divergence problems (like unrealistic low pressures).
Do you think interpreting the udf makes this problem? if I compile the udf, would it be solved?
What is the basic difference between interpreting and compiling a udf?

If there are no warning messages while solving the case with the interpreted UDF, then there should not be any difference when you compile the UDF instead.

In general, interpreted UDFs are the "quick and dirty" solution if the UDF doesn't contain complicated calculations.
There are some restrictions to interpreted UDFs concerning both the C-programming and the data read from the solver. Refer to the UDF manual for details.
A compiled UDF does not have these restrictions and runs much faster.

In your case, it is far more likely that the content of the UDF itself causes the instability of the solving process.